Polya question

I know this a little off from tuning - but I had a question about
Polya's polynomial question that Gene or someone might be able to
answer.

Using the normal technique, for example, generates 5 symmetrical triads,
with this grid:

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
0 10 0 10 0 10 0 10 0 10 0 10 = 60 / 12 = 5

Common sense tells me it should be:

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
0 0 0 4 0 0 0 4 0 10 0 52 = 60 / 12 = 5

I can't find the formula for "my" way. This is the only place in Polya's
polynomial method where things kind of break down. Even with the
complementability problem (Gilbert and Riordan, 1961) everything pans
out beautifully - for example, with three colors in 6-bead necklaces
the expansion for T3 works out to (2 + 1)^3 which is 2^3 + 3(2^2*1)+ 3
(2^1*1^2)+ 1^3. Each part of the polynomial corresponds to an exact
type of necklace, in this case 3-3, 2-2-2, 4-1-1, 6-0 respectively.
Quite nice.

I know my necklace posts haven't been exactly popular, except with
maybe Jon Wild. Michael Keith actually does use the 3-color problem
in "From Polynomials to Polya" to define metanecklaces that
are used to calculate adjacencies, max interval, span and interval
types. I hope to find a use for complementable metanecklaces.

Since I've already outworn my welcome, here is the same discrepency
between Polya and my method for symmetrical hexachords:

Polya:

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
20 20 20 20 20 20 20 20 20 20 20 20= 240 / 12 = 20

Common sense tells me it should be:

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
0 2 0 6 0 8 0 6 0 2 0 216 / 12 = 20

Both ways produce the same result, and since I can't find a formula for
my way, of course, I use Polya's. I just don't understand why his
method doesn't produce the correct sub-results for transposes.

The part for transposes alone, does (without reducing for mirror-image)

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
0 2 0 6 0 20 0 6 0 2 0 924 / 12 = 80

Perfect.

My concern also lies in what happens when you start to calculate
for D4 X S3, etc. Do these have the right transpositional
representations?

Thanks

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> I know this a little off from tuning - but I had a question about
> Polya's polynomial question that Gene or someone might be able to
> answer.
>
> Using the normal technique, for example, generates 5 symmetrical
triads,
> with this grid:
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 0 10 0 10 0 10 0 10 0 10 0 10 = 60 / 12 = 5
>
> Common sense tells me it should be:
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 0 0 0 4 0 0 0 4 0 10 0 52 = 60 / 12 = 5
>
> I can't find the formula for "my" way. This is the only place in
Polya's
> polynomial method where things kind of break down. Even with the
> complementability problem (Gilbert and Riordan, 1961) everything
pans
> out beautifully - for example, with three colors in 6-bead
necklaces
> the expansion for T3 works out to (2 + 1)^3 which is 2^3 + 3(2^2*1)
+ 3
> (2^1*1^2)+ 1^3. Each part of the polynomial corresponds to an exact
> type of necklace, in this case 3-3, 2-2-2, 4-1-1, 6-0 respectively.
> Quite nice.
>
> I know my necklace posts haven't been exactly popular, except with
> maybe Jon Wild. Michael Keith actually does use the 3-color problem
> in "From Polynomials to Polya" to define metanecklaces that
> are used to calculate adjacencies, max interval, span and interval
> types. I hope to find a use for complementable metanecklaces.
>
> Since I've already outworn my welcome, here is the same discrepency
> between Polya and my method for symmetrical hexachords:
>
> Polya:
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 20 20 20 20 20 20 20 20 20 20 20 20= 240 / 12 = 20
>
> Common sense tells me it should be:
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 0 2 0 6 0 8 0 6 0 2 0 216 / 12 = 20
>
> Both ways produce the same result, and since I can't find a formula
for
> my way, of course, I use Polya's. I just don't understand why his
> method doesn't produce the correct sub-results for transposes.
>
> The part for transposes alone, does (without reducing for mirror-
image)
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 0 2 0 6 0 20 0 6 0 2 0 924 / 12 = 80
>
> Perfect.
>
> My concern also lies in what happens when you start to calculate
> for D4 X S3, etc. Do these have the right transpositional
> representations?
>
> Thanks
>

The "actual" grid for symmetrical sets is:

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12/T0
2 4 8 16 2 52 2 16 8 4 2 1036 Sum=1152 /12 =96

This is in contrast with the Polya grid:

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12/T0
64 128 64 128 64 128 64 128 64 128 64 128 Sum=1152 /12 = 96

This is because you get:

(x^2+1)^6 for odd transposes, which adds to 64
(x^2+1)^5(x+1)^2 for even tranposes, which adds to 128

Expanding out the polynomials gives, for example, "20" for every
transpose of hexachords, but the question is, is that "legal"?

Paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> I know this a little off from tuning - but I had a question about
> Polya's polynomial question that Gene or someone might be able to
> answer.

It would help if you reminded us what the notation is. What does Tn
stand for?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
> >
> > I know this a little off from tuning - but I had a question about
> > Polya's polynomial question that Gene or someone might be able to
> > answer.
>
> It would help if you reminded us what the notation is. What does Tn
> stand for?
>
Sorry, Tn is tranposition by n displacement. In the case of my Polya
grids, it is sets that map into themselves through Tn. In the case
of inverse complementation, it is sets that map into themselves through
TnI (transpose + mirror inversion). One thing I am not settled on,
is should it be transpose THEN inverse or inverse THEN tranpose?

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> I know this a little off from tuning - but I had a question about
> Polya's polynomial question that Gene or someone might be able to
> answer.
>
> Using the normal technique, for example, generates 5 symmetrical
triads,
> with this grid:
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 0 10 0 10 0 10 0 10 0 10 0 10 = 60 / 12 = 5
>
> Common sense tells me it should be:
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 0 0 0 4 0 0 0 4 0 10 0 52 = 60 / 12 = 5
>
> I can't find the formula for "my" way. This is the only place in
Polya's
> polynomial method where things kind of break down. Even with the
> complementability problem (Gilbert and Riordan, 1961) everything
pans
> out beautifully - for example, with three colors in 6-bead
necklaces
> the expansion for T3 works out to (2 + 1)^3 which is 2^3 + 3(2^2*1)
+ 3
> (2^1*1^2)+ 1^3. Each part of the polynomial corresponds to an exact
> type of necklace, in this case 3-3, 2-2-2, 4-1-1, 6-0 respectively.
> Quite nice.
>
> I know my necklace posts haven't been exactly popular, except with
> maybe Jon Wild. Michael Keith actually does use the 3-color problem
> in "From Polynomials to Polya" to define metanecklaces that
> are used to calculate adjacencies, max interval, span and interval
> types. I hope to find a use for complementable metanecklaces.
>
> Since I've already outworn my welcome, here is the same discrepency
> between Polya and my method for symmetrical hexachords:
>
> Polya:
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 20 20 20 20 20 20 20 20 20 20 20 20= 240 / 12 = 20
>
> Common sense tells me it should be:
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 0 2 0 6 0 8 0 6 0 2 0 216 / 12 = 20
>
> Both ways produce the same result, and since I can't find a formula
for
> my way, of course, I use Polya's. I just don't understand why his
> method doesn't produce the correct sub-results for transposes.
>
> The part for transposes alone, does (without reducing for mirror-
image)
>
> T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
> 0 2 0 6 0 20 0 6 0 2 0 924 / 12 = 80
>
> Perfect.
>
> My concern also lies in what happens when you start to calculate
> for D4 X S3, etc. Do these have the right transpositional
> representations?
>
> Thanks

Found the error in my thinking. Symmetrical sets map to themselves
under TnI, not Tn, so the grids I have presented have real limited
value. The good news - the Polya method for symmetrical sets is
consistent in terms of transpositions under TnI. So I trust it will
be so for D4 X S3 as well. Just for fun, I will create a spreadsheet
with T1-T12 as columns and cardinality 1-12 as rows, (Sets from 1 -
12 members that map into themselves under D4 X S3 = Tn/TnI

Paul Hj